Properties

Label 2-28e2-1.1-c3-0-14
Degree $2$
Conductor $784$
Sign $1$
Analytic cond. $46.2574$
Root an. cond. $6.80128$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·3-s + 8·5-s + 73·9-s + 40·11-s + 12·13-s − 80·15-s + 58·17-s + 26·19-s + 64·23-s − 61·25-s − 460·27-s − 62·29-s + 252·31-s − 400·33-s + 26·37-s − 120·39-s − 6·41-s − 416·43-s + 584·45-s − 396·47-s − 580·51-s − 450·53-s + 320·55-s − 260·57-s + 274·59-s + 576·61-s + 96·65-s + ⋯
L(s)  = 1  − 1.92·3-s + 0.715·5-s + 2.70·9-s + 1.09·11-s + 0.256·13-s − 1.37·15-s + 0.827·17-s + 0.313·19-s + 0.580·23-s − 0.487·25-s − 3.27·27-s − 0.397·29-s + 1.46·31-s − 2.11·33-s + 0.115·37-s − 0.492·39-s − 0.0228·41-s − 1.47·43-s + 1.93·45-s − 1.22·47-s − 1.59·51-s − 1.16·53-s + 0.784·55-s − 0.604·57-s + 0.604·59-s + 1.20·61-s + 0.183·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(46.2574\)
Root analytic conductor: \(6.80128\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{784} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.376159670\)
\(L(\frac12)\) \(\approx\) \(1.376159670\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + 10 T + p^{3} T^{2} \)
5 \( 1 - 8 T + p^{3} T^{2} \)
11 \( 1 - 40 T + p^{3} T^{2} \)
13 \( 1 - 12 T + p^{3} T^{2} \)
17 \( 1 - 58 T + p^{3} T^{2} \)
19 \( 1 - 26 T + p^{3} T^{2} \)
23 \( 1 - 64 T + p^{3} T^{2} \)
29 \( 1 + 62 T + p^{3} T^{2} \)
31 \( 1 - 252 T + p^{3} T^{2} \)
37 \( 1 - 26 T + p^{3} T^{2} \)
41 \( 1 + 6 T + p^{3} T^{2} \)
43 \( 1 + 416 T + p^{3} T^{2} \)
47 \( 1 + 396 T + p^{3} T^{2} \)
53 \( 1 + 450 T + p^{3} T^{2} \)
59 \( 1 - 274 T + p^{3} T^{2} \)
61 \( 1 - 576 T + p^{3} T^{2} \)
67 \( 1 - 476 T + p^{3} T^{2} \)
71 \( 1 - 448 T + p^{3} T^{2} \)
73 \( 1 - 158 T + p^{3} T^{2} \)
79 \( 1 - 936 T + p^{3} T^{2} \)
83 \( 1 - 530 T + p^{3} T^{2} \)
89 \( 1 - 390 T + p^{3} T^{2} \)
97 \( 1 + 214 T + p^{3} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.883874733187806805399171449028, −9.571763422226582666139710388941, −8.056644876850605934346586893490, −6.79331358184166884700235923981, −6.38839091488719359301234328981, −5.51604038535645283303238773729, −4.81860036652169941072925078794, −3.65884307970003338983827350635, −1.66676436429097790343681844714, −0.78687624371832426228226687542, 0.78687624371832426228226687542, 1.66676436429097790343681844714, 3.65884307970003338983827350635, 4.81860036652169941072925078794, 5.51604038535645283303238773729, 6.38839091488719359301234328981, 6.79331358184166884700235923981, 8.056644876850605934346586893490, 9.571763422226582666139710388941, 9.883874733187806805399171449028

Graph of the $Z$-function along the critical line